STANFORD UNIV CALIF Dept. of MATHEMATICS
Overview
Works:  31 works in 31 publications in 1 language and 31 library holdings 

Classifications:  QA273, 519.1 
Publication Timeline
.
Most widely held works by
STANFORD UNIV CALIF Dept. of MATHEMATICS
The effective conductivity of a regular composite medium by
Stanford University(
Book
)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
The advantages of a Green's function method are exemplified by the analysis of steady potential flow through a homogeneous medium containing a regular array of subdomains where a different value of the conductivity prevails. It is shown, in particular, that the results previously obtained by Rayleigh for the square array of cylindrical obstacles with infinite conductivity are recovered in a more efficient manner; and that the method is suitable for handling either widely dispersed or closely packed arrays. (Author)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
The advantages of a Green's function method are exemplified by the analysis of steady potential flow through a homogeneous medium containing a regular array of subdomains where a different value of the conductivity prevails. It is shown, in particular, that the results previously obtained by Rayleigh for the square array of cylindrical obstacles with infinite conductivity are recovered in a more efficient manner; and that the method is suitable for handling either widely dispersed or closely packed arrays. (Author)
On the theory of Kirchhoff's method for the determination of electrical conductivity by
Stanford University(
Book
)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
A measure of unification is achieved for the purpose of analyzing the steady current flows through elongated cylinders of general section. These are activated by a pair of thin electrodes on the end faces and are relevant to a scheme of Kirchhoff for determining the electrical conductivity of homogeneous materials, and is manifest by the central role of a harmonic Green's function which satisfies a Neumann condition on the boundary of an infinitely long duct whose section duplicates that of the appropriate cylinder. The utility of a Green's function formalism is brought out with specific numerical details in the respective cases of circular and square cylinder section. (Author)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
A measure of unification is achieved for the purpose of analyzing the steady current flows through elongated cylinders of general section. These are activated by a pair of thin electrodes on the end faces and are relevant to a scheme of Kirchhoff for determining the electrical conductivity of homogeneous materials, and is manifest by the central role of a harmonic Green's function which satisfies a Neumann condition on the boundary of an infinitely long duct whose section duplicates that of the appropriate cylinder. The utility of a Green's function formalism is brought out with specific numerical details in the respective cases of circular and square cylinder section. (Author)
Scattering of surface waves by submerged circular cylinders. Part II: Scattering by an infinite array of cylinders by
Stanford University(
Book
)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
The paper deals with progressive surface wave motions of a small amplitude and timeperiodic nature in an ideal fluid which contains an infinite submerged array of parallel and identical circular cylinders. Their characterization is sought in the framework of a homogeneous linear boundary value problem for harmonic functions which exhibit a common phase shift (that plays the role of an eigenvalue) for any displacement along the direction of wave propagation with a magnitude equal to the separation between adjacent cylinders. An equivalent integral equation is formulated and reduced to an infinite homogeneous linear system from which there obtains a determinantal condition that specifies the dependence of the eigenvalue, or effective wave number, on the frequency, gravitation constant, and geometric parameters. Approximations to the eigenvalue are calculated for any fixed wave length when (1) the radius of the cylinders is small compared with their spacing or (2) their separation distance is large with respect to the radius and depth of immersion. (Author)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
The paper deals with progressive surface wave motions of a small amplitude and timeperiodic nature in an ideal fluid which contains an infinite submerged array of parallel and identical circular cylinders. Their characterization is sought in the framework of a homogeneous linear boundary value problem for harmonic functions which exhibit a common phase shift (that plays the role of an eigenvalue) for any displacement along the direction of wave propagation with a magnitude equal to the separation between adjacent cylinders. An equivalent integral equation is formulated and reduced to an infinite homogeneous linear system from which there obtains a determinantal condition that specifies the dependence of the eigenvalue, or effective wave number, on the frequency, gravitation constant, and geometric parameters. Approximations to the eigenvalue are calculated for any fixed wave length when (1) the radius of the cylinders is small compared with their spacing or (2) their separation distance is large with respect to the radius and depth of immersion. (Author)
On a procedure for analyzing certain problems of diffusion theory by
Stanford University(
Book
)
1 edition published in 1972 in English and held by 1 WorldCat member library worldwide
A theoretical procedure is described for resolving initialboundary value problems relative to partial differential equations of the diffusion type in cases of axial symmetry. This entails a single integral transformation or series expansion with Bessel functions depending on the radial coordinate and uses a full solution of the reduced form of the diffusion equation involving the axial coordinate and the time. Applications are made to problems of heat conduction, electrical conductivity and viscous fluid motion. (Author)
1 edition published in 1972 in English and held by 1 WorldCat member library worldwide
A theoretical procedure is described for resolving initialboundary value problems relative to partial differential equations of the diffusion type in cases of axial symmetry. This entails a single integral transformation or series expansion with Bessel functions depending on the radial coordinate and uses a full solution of the reduced form of the diffusion equation involving the axial coordinate and the time. Applications are made to problems of heat conduction, electrical conductivity and viscous fluid motion. (Author)
Lectures on boundary theory for Markov chains by
Kai Lai Chung(
Book
)
1 edition published in 1971 in English and held by 1 WorldCat member library worldwide
The boundary theory as viewed in this book is a study of Markov chains with certain infinities (discontinuities that are not jumps) in the sample functions. In the first part a simple 'boundary' is set up by a direct, intuitive method and the basic relevant sample behavior is studied. The second, more substantial part of this book deals with the analysis of the chain in relation to its menimal part and its exits and entrances to and from the boundary. (Author)
1 edition published in 1971 in English and held by 1 WorldCat member library worldwide
The boundary theory as viewed in this book is a study of Markov chains with certain infinities (discontinuities that are not jumps) in the sample functions. In the first part a simple 'boundary' is set up by a direct, intuitive method and the basic relevant sample behavior is studied. The second, more substantial part of this book deals with the analysis of the chain in relation to its menimal part and its exits and entrances to and from the boundary. (Author)
Asymptotic distribution of generalized integers and prime numbers by
Harold G Diamond(
Book
)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
The paper is divided into four chapters, of which the first two are concerned with a wide class of measures and general properties of these measures. The last two chapters deal with a few measures of prime number theoretic interest, and for these measures some special results are proved. In the first chapter 'C' and 'R' classes of measures are introduced. Convolution and certain other operations are defined and their elementary properties explored. A topology is induced in 'C' by total variation semi norms, and 'C' is shown to be a complete topological group under addition. The second chapter begins with some general results about convergence and continuity of power series in measures. C sub 1 denotes the elements of 'C' with point mass at 1, and all measures in C sub 1 is shown to be invertible in a convolution sense. The main result of the chapter is the representation of C sub 1 by 'C' under the exponential mapping. This mapping is shown to be a homeomorphism and an isomorphism of topological groups. As a corollary to this theorem the existence of nth roots of measures in C sub 1 is infered. Gprimes and integers and their related measures are defined in the third chapter. Using the results of Chapter 2, the counting measure of gintegers and Lebesgue measure is shown to be given as exponentials. The prime number theorem for gprimes is the main result of the fourth chapter
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
The paper is divided into four chapters, of which the first two are concerned with a wide class of measures and general properties of these measures. The last two chapters deal with a few measures of prime number theoretic interest, and for these measures some special results are proved. In the first chapter 'C' and 'R' classes of measures are introduced. Convolution and certain other operations are defined and their elementary properties explored. A topology is induced in 'C' by total variation semi norms, and 'C' is shown to be a complete topological group under addition. The second chapter begins with some general results about convergence and continuity of power series in measures. C sub 1 denotes the elements of 'C' with point mass at 1, and all measures in C sub 1 is shown to be invertible in a convolution sense. The main result of the chapter is the representation of C sub 1 by 'C' under the exponential mapping. This mapping is shown to be a homeomorphism and an isomorphism of topological groups. As a corollary to this theorem the existence of nth roots of measures in C sub 1 is infered. Gprimes and integers and their related measures are defined in the third chapter. Using the results of Chapter 2, the counting measure of gintegers and Lebesgue measure is shown to be given as exponentials. The prime number theorem for gprimes is the main result of the fourth chapter
Research in the theory of riemann surfaces(
Book
)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
The report presents a final summary of and a listing of reports generated under, a study of Riemann surfaces; the relationship between differentialgeometric and functiontheoretic properties, with applications and extensions to minimal surfaces and Riemannian manifolds; a study of the moduli of Riemann surfaces; quasiconformal mappings between Riemann surfaces; the variational theory of Riemann surfaces; and applications of abstract analysis to the theory of Riemann surfaces. (Author)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
The report presents a final summary of and a listing of reports generated under, a study of Riemann surfaces; the relationship between differentialgeometric and functiontheoretic properties, with applications and extensions to minimal surfaces and Riemannian manifolds; a study of the moduli of Riemann surfaces; quasiconformal mappings between Riemann surfaces; the variational theory of Riemann surfaces; and applications of abstract analysis to the theory of Riemann surfaces. (Author)
On the problem of closely separated circular discs at equal potential by
Stanford University(
Book
)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
A pair of identical and coaxially aligned circular discs is maintained at unit potential, inducing a common axisymmetric charge distribution on their respective surfaces. An approximate solution of the related boundary value problem is sought when the ratio epsilon of separation against diameter is small, with particular emphasis given to the asymptotic development of the total charge C on each disc. A crude estimate therefore, C approximately equal to 1/pi, obtains readily on the basis of an intuitive argument which identifies the potential field for narrowly separated discs with that of a coincident pair and assigns to each onehalf of the density (or total charge) found on the limiting surface at the assigned potential. More precise results are obtained, firstly, by using the particulars of the single disc field distribution in a complementary pair of variational principles, and secondly, by relying on a uniformly accurate characterization of the potential over the whole plane containing one of the discs, which evidences different behaviors for points close to and far from the edge, respectively
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
A pair of identical and coaxially aligned circular discs is maintained at unit potential, inducing a common axisymmetric charge distribution on their respective surfaces. An approximate solution of the related boundary value problem is sought when the ratio epsilon of separation against diameter is small, with particular emphasis given to the asymptotic development of the total charge C on each disc. A crude estimate therefore, C approximately equal to 1/pi, obtains readily on the basis of an intuitive argument which identifies the potential field for narrowly separated discs with that of a coincident pair and assigns to each onehalf of the density (or total charge) found on the limiting surface at the assigned potential. More precise results are obtained, firstly, by using the particulars of the single disc field distribution in a complementary pair of variational principles, and secondly, by relying on a uniformly accurate characterization of the potential over the whole plane containing one of the discs, which evidences different behaviors for points close to and far from the edge, respectively
A note on problems of wave generation in semiinfinite media by surface forces by
Stanford University(
Book
)
1 edition published in 1972 in English and held by 1 WorldCat member library worldwide
A classical problem of unsteady wave excitation, which envisages loading or primary sources on the boundary of a halfspace, is resolved in a direct and efficient fashion, yielding various theoretical representations of the solution and unifying previously reported analyses. (Author)
1 edition published in 1972 in English and held by 1 WorldCat member library worldwide
A classical problem of unsteady wave excitation, which envisages loading or primary sources on the boundary of a halfspace, is resolved in a direct and efficient fashion, yielding various theoretical representations of the solution and unifying previously reported analyses. (Author)
Acoustical diffraction radiation by
Stanford University(
Book
)
1 edition published in 1971 in English and held by 1 WorldCat member library worldwide
A finite region of turbulent flow is represented by a volume distribution of quadrupolar sources, according to the Lighthill model, for the purpose of estimating the sound level generated in the surrounding region. By way of incorporating environmental features into such a model of free turbulence it becomes appropriate to examine quadrupolar source interaction with material surfaces; this poses a formidable diffraction or scattering problem unless the surface is compact on the acoustic wave length scale or of simple geometrical shape. The calculations by Ffowcs Williams and Hall (1970), Crighton and Leppington (1971) relative to an immobile point quadrupole near the straight edge of a thin semiinfinite plane screen, which bring out the possibility of a substantial amplification in the far field as the result of scattering by the screen, suggest a related rise for the output level of an actual flow in a comparable setting; and they prompt the consideration of acoustical effects attributable to the motion of sources past the edge. (Author)
1 edition published in 1971 in English and held by 1 WorldCat member library worldwide
A finite region of turbulent flow is represented by a volume distribution of quadrupolar sources, according to the Lighthill model, for the purpose of estimating the sound level generated in the surrounding region. By way of incorporating environmental features into such a model of free turbulence it becomes appropriate to examine quadrupolar source interaction with material surfaces; this poses a formidable diffraction or scattering problem unless the surface is compact on the acoustic wave length scale or of simple geometrical shape. The calculations by Ffowcs Williams and Hall (1970), Crighton and Leppington (1971) relative to an immobile point quadrupole near the straight edge of a thin semiinfinite plane screen, which bring out the possibility of a substantial amplification in the far field as the result of scattering by the screen, suggest a related rise for the output level of an actual flow in a comparable setting; and they prompt the consideration of acoustical effects attributable to the motion of sources past the edge. (Author)
Scattering of xrays by layered structures by William Randolph Jones(
Book
)
1 edition published in 1966 in English and held by 1 WorldCat member library worldwide
The diffraction of Xrays by perfect layered crystal structures is analyzed, with particular attention given to both the incidence of plane waves and collimated beams on a crystal plate of finite breadth set for Laue diffraction (or substantial transmission). The plane wave diffraction effects are obtained by a mode of analysis employing integral equations and the WienerHopf method of solution; and it is shown that the analytical expressions for the transmission coefficients agree, as regards their leading terms, with the results of the two wave approximation for the diffraction theory of periodic structures derived by von Laue and others. The Xray beam analysis, based on the principle of superposition of plane wave components, is developed to yield directly the spatial profiles of the transmitted beams and the phenomenon of beam separation; a significant outcome of the present approach is that the calculated profiles remain finite at the beam edges, in contrast with the singular behavior exhibited by reviously given distributions. (Author)
1 edition published in 1966 in English and held by 1 WorldCat member library worldwide
The diffraction of Xrays by perfect layered crystal structures is analyzed, with particular attention given to both the incidence of plane waves and collimated beams on a crystal plate of finite breadth set for Laue diffraction (or substantial transmission). The plane wave diffraction effects are obtained by a mode of analysis employing integral equations and the WienerHopf method of solution; and it is shown that the analytical expressions for the transmission coefficients agree, as regards their leading terms, with the results of the two wave approximation for the diffraction theory of periodic structures derived by von Laue and others. The Xray beam analysis, based on the principle of superposition of plane wave components, is developed to yield directly the spatial profiles of the transmitted beams and the phenomenon of beam separation; a significant outcome of the present approach is that the calculated profiles remain finite at the beam edges, in contrast with the singular behavior exhibited by reviously given distributions. (Author)
Numerical Calculation of GravityCapillary Interfacial Waves of Finite Amplitude(
Book
)
1 edition published in 1980 in English and held by 1 WorldCat member library worldwide
1 edition published in 1980 in English and held by 1 WorldCat member library worldwide
Decision procedures for real and padic fields by
Paul J Cohen(
Book
)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
A decision procedure is given for the real number field which reproves a result of Tarski, and for padic fields a procedure is given to reduce problems about the field to statements concerning the residue class rings. This gives a purely effective proof of the recent results of Ax and Kochen. The methods used point up the similarity of the two cases. Thus whereas Sturm's theorem can be used in the real case, our proof yields an inductive procedure for finding roots in the padic case. Finally, the application to Artin's conjecture is discussed and it is shown that the exceptional primes are primitive recursive functions of the degree. (Author)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
A decision procedure is given for the real number field which reproves a result of Tarski, and for padic fields a procedure is given to reduce problems about the field to statements concerning the residue class rings. This gives a purely effective proof of the recent results of Ax and Kochen. The methods used point up the similarity of the two cases. Thus whereas Sturm's theorem can be used in the real case, our proof yields an inductive procedure for finding roots in the padic case. Finally, the application to Artin's conjecture is discussed and it is shown that the exceptional primes are primitive recursive functions of the degree. (Author)
Scattering of surface waves by submerged circular cylinders. part i: scattering by two cylinders by
Stanford University(
Book
)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
The paper is concerned with the scattering of small amplitude waves on the surface of an incompressible and inviscid fluid in which a pair of rigid and parallel circular cylinders are wholly immersed. Attention is given to the setting wherein a straightcrested and timeperiodic primary wave train advances in a direction normal to the cylinder axes and, in particular, to the characterization of the accompanying reflection and transmission coefficients for the secondary wave trains. If the plane or linearized form of the free surface condition be adopted, resolution of the harmonic boundary value problem thus posed is shown to be equivalent to solving an infinite system of linear equations, and this equivalence is also utilized for theoretical considerations bearing on the existence and uniqueness of solutions. The analysis forms an extension of others which have previously established the absence of reflection for a single submerged cylinder in similar circumstances; and reveals that the latter feature does not carry over to the twocylinder configuration, although the amount of reflection becomes arbitrarily small when the cylinders are sufficiently far apart. (Author)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
The paper is concerned with the scattering of small amplitude waves on the surface of an incompressible and inviscid fluid in which a pair of rigid and parallel circular cylinders are wholly immersed. Attention is given to the setting wherein a straightcrested and timeperiodic primary wave train advances in a direction normal to the cylinder axes and, in particular, to the characterization of the accompanying reflection and transmission coefficients for the secondary wave trains. If the plane or linearized form of the free surface condition be adopted, resolution of the harmonic boundary value problem thus posed is shown to be equivalent to solving an infinite system of linear equations, and this equivalence is also utilized for theoretical considerations bearing on the existence and uniqueness of solutions. The analysis forms an extension of others which have previously established the absence of reflection for a single submerged cylinder in similar circumstances; and reveals that the latter feature does not carry over to the twocylinder configuration, although the amount of reflection becomes arbitrarily small when the cylinders are sufficiently far apart. (Author)
Intercalation theorems for pushdown store and stack languages by W. F Ogden(
Book
)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
The thesis develops necessary conditions for languages to be contextfree, deterministic contextfree, stack generable, stack decidable, and nonerasing stack generable. The result for contextfree languages is a generalization of the well known necessary condition of BarHillel et al., but it does provide an alternative approach to inherent ambiguity in contextfree languages. (Author)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
The thesis develops necessary conditions for languages to be contextfree, deterministic contextfree, stack generable, stack decidable, and nonerasing stack generable. The result for contextfree languages is a generalization of the well known necessary condition of BarHillel et al., but it does provide an alternative approach to inherent ambiguity in contextfree languages. (Author)
Estimates for the transfinite diameter of a continuum(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
Some estimates for the transfinite diameter of a continuum are derived in terms of elementary geometric quantities connected with it. The main intention is methodological. Variational methods are applied, and the corresponding extremum continua are described by a functional differential equation. In all cases the determination of the extremum continuum is based on rather simple geometric arguments. It is intended that the arguments used should indicate more general methods of solution which could be applied in the calculus of variations for conformal mappings. (Author)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
Some estimates for the transfinite diameter of a continuum are derived in terms of elementary geometric quantities connected with it. The main intention is methodological. Variational methods are applied, and the corresponding extremum continua are described by a functional differential equation. In all cases the determination of the extremum continuum is based on rather simple geometric arguments. It is intended that the arguments used should indicate more general methods of solution which could be applied in the calculus of variations for conformal mappings. (Author)
Bubble or Drop Distortion in a Straining Flow in Two Dimensions(
Book
)
1 edition published in 1980 in English and held by 1 WorldCat member library worldwide
1 edition published in 1980 in English and held by 1 WorldCat member library worldwide
Reflection from a periodically composed halfspace by
Stanford University(
Book
)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
The character of wave motions in geometrical settings of a periodic nature and their interplay with primary and secondary waves in the exterior of partly spacefilling configurations receive prominence in the theories of xray and electron diffraction. It is customary to focus attention on the free or natural wave motions in unbounded media with a periodic bias prior to establishing the selfconsistent solution for specific excitations in composite media. An integrated, rather than sequential, approach to problems in the latter category is recommended and detailed here in connection with the incidence of plane waves on a halfspace with periodic composition normal to its boundary; and is shown, in particular, to furnish directly a superior means of calculating the reflection coefficient. (Author)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
The character of wave motions in geometrical settings of a periodic nature and their interplay with primary and secondary waves in the exterior of partly spacefilling configurations receive prominence in the theories of xray and electron diffraction. It is customary to focus attention on the free or natural wave motions in unbounded media with a periodic bias prior to establishing the selfconsistent solution for specific excitations in composite media. An integrated, rather than sequential, approach to problems in the latter category is recommended and detailed here in connection with the incidence of plane waves on a halfspace with periodic composition normal to its boundary; and is shown, in particular, to furnish directly a superior means of calculating the reflection coefficient. (Author)
Quantum Field Theory, Differential Operators and Scattering Theory(
Book
)
1 edition published in 1971 in English and held by 1 WorldCat member library worldwide
The research of this contract has covered several areas, including differential equations, harmonic analysis, functional analysis, logic, and quantum field theory. Some work has been done on the theoretical foundations of field theory while other work has been done on the classical problems in scattering theory arising from the physical situations. In harmonic analysis, questions about function spaces of analytic functions in more than one variable and classical problems in potential theory have been investigated. Tauberian theorems in their relation to number theory, and the general scope of the analytic methods of harmonic analysis in the theory of distribution of prime numbers have been considered. In Logic problems in the existence of models with automorphisms as well as general decision problems were investigated. Some recent results concern the application of automata in deciding various second order theories. Also work was done which offers hope of giving constructive estimates for problems in Diophantine Equations for which only nonconstructive methods now exist. (Author)
1 edition published in 1971 in English and held by 1 WorldCat member library worldwide
The research of this contract has covered several areas, including differential equations, harmonic analysis, functional analysis, logic, and quantum field theory. Some work has been done on the theoretical foundations of field theory while other work has been done on the classical problems in scattering theory arising from the physical situations. In harmonic analysis, questions about function spaces of analytic functions in more than one variable and classical problems in potential theory have been investigated. Tauberian theorems in their relation to number theory, and the general scope of the analytic methods of harmonic analysis in the theory of distribution of prime numbers have been considered. In Logic problems in the existence of models with automorphisms as well as general decision problems were investigated. Some recent results concern the application of automata in deciding various second order theories. Also work was done which offers hope of giving constructive estimates for problems in Diophantine Equations for which only nonconstructive methods now exist. (Author)
Reflection and Transmission at a Plane Screen with Periodically Arranged Circular or Elliptical Apertures by
Stanford University(
Book
)
1 edition published in 1972 in English and held by 1 WorldCat member library worldwide
Theoretical estimates are given for the scattered acoustic wave amplitudes which result if a plane wave is incident on a perforated screen containing a periodic array of circular or elliptical apertures, on the twin assumptions that: (1) the wave length greatly exceeds the aperture spacings and (2) the aperture dimensions are small compared to the spacings. The effect of a parallel backing plane is described and the impedance of a plane with a regular array of baffled pistons is calculated. (Author)
1 edition published in 1972 in English and held by 1 WorldCat member library worldwide
Theoretical estimates are given for the scattered acoustic wave amplitudes which result if a plane wave is incident on a perforated screen containing a periodic array of circular or elliptical apertures, on the twin assumptions that: (1) the wave length greatly exceeds the aperture spacings and (2) the aperture dimensions are small compared to the spacings. The effect of a parallel backing plane is described and the impedance of a plane with a regular array of baffled pistons is calculated. (Author)
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